10.
The Reel Good Cinema is conducting a mathematical study. In its theater, there are
200 seats. Adult tickets cost $12.50 and child tickets cost $6.25. The cinema's goal is
to sell at least $1500 worth of tickets for the theater.
Write a system of linear inequalities that can be used to find the possible combinations
of adult tickets, x, and child tickets, y, that would satisfy the cinema's goal.
Graph the solution to this system of inequalities on the set of axes below. Label the
solution with an S.
Marta claims that selling 30 adult tickets and 80 child tickets will result in meeting the
cinema's goal. Explain whether she is correct or incorrect, based on the graph drawn.

A. To answer this question, we must consider two inequalities. The first inequality we will construct is regarding the number of tickets sold with respect to capacity. The theater cannot exceed the allotted 200 seats, so this would indicate we would use a less than or equal to inequality. The inequality we would construct to illustrate this using adult and child tickets is x + y <= 200.
The second inequality we will construct is based upon the ticket prices set and the theater’s goal. The problem indicates the theater wants to sell “at least $1,500 worth of tickets...” To create the inequality here, we will create it as follows: 12.5x + 6.25y >= 1,500.
To find the intersection point, we must isolate y. In order to do this, move x to the other side of both inequalities. Inequality 2 can be simplified to y >= -12.5x / 6.25 + 1,500 / 6.25. Graph and identify the solution from there.